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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '128.2220', 'ambient_counter': 2220, 'ambient_order': 128, 'ambient_tex': 'D_4^2:C_2', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 32, 'counter': 40, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '128.2220.4.bb1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.bb1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '32.28', 'subgroup_hash': 28, 'subgroup_order': 32, 'subgroup_tex': 'C_4:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.2220', 'aut_centralizer_order': None, 'aut_label': '4.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '16.e1', 'complements': ['32.u1', '32.v1'], 'conjugacy_class_count': 3, 'contained_in': ['2.a1', '2.g1'], 'contains': ['8.o1', '8.s1', '8.u1', '8.w1', '8.bf1'], 'core': '4.bb1', 'coset_action_label': None, 'count': 3, 'diagramx': [3806, 9424, 3813, 9495], 'generators': [95252536, 132452760, 160762338, 1861343, 16], 'label': '128.2220.4.bb1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.bb1', 'normal_contained_in': ['2.a1', '2.g1'], 'normal_contains': ['8.o1', '8.s1', '8.u1', '8.w1', '8.bf1'], 'normalizer': '1.a1', 'old_label': '4.bb1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.bb1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 8], [17, 2, 8], [5, 18, 8], [17, 18, 12], [5, 6, 24], [1, 6, 8], [1, 14, 8]], 'aut_group': '128.2163', 'aut_hash': 2163, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 12, 'aut_perms': [101693656, 7, 182610840, 14923560, 138989520, 248266207], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 4, 2, 1], [4, 2, 2, 2], [4, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 2], [4, 2, 4], [4, 4, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 28, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 6, 8], [5, 2, 24], [1, 18, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.28', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4:D4', 'ngens': 3, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 14, 'number_divisions': 13, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 47, 'number_subgroups': 74, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [8, 0, 0, 0], 'outer_gens': [[1, 10, 8], [1, 2, 12], [1, 2, 24], [17, 2, 8]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 1]], 'representations': {'PC': {'code': 604570138, 'gens': [1, 2, 4], 'pres': [5, 2, 2, 2, 2, 2, 61, 26, 248, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931074, 26128940, 26483560]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 29555524, 28816400, 28233362, 33533700]}, 'Perm': {'d': 8, 'gens': [1567, 13, 5183, 23, 11520]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4:D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2], 'aut_gens': [[41455447, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455463, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 125199496, 5, 247258080, 95252536, 132452760, 16], [171682703, 125199480, 5, 41091256, 95252536, 132452760, 16], [94525343, 247258080, 5, 41091240, 95252536, 132452760, 16], [171682703, 11290320, 132452765, 172046896, 301245976, 132452760, 16], [41455447, 11290320, 5, 41091240, 95252536, 132452760, 16], [41455447, 11290320, 14, 41091240, 95252520, 132452760, 16], [171682703, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 11290336, 132452765, 41091256, 95252536, 132452760, 16], [262897943, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 11290336, 14, 41091256, 95252536, 132452760, 16], [171682703, 125199480, 5, 172046880, 95252536, 132452760, 16], [41455463, 11290320, 5, 41091256, 95252536, 132452760, 16]], 'aut_group': '12288.bmj', 'aut_hash': 4169318348325865856, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 20, 'aut_perms': [92892238207471080, 1488344784099308755, 1709812413405338465, 2339819056632805314, 1712324462084256335, 701180240859491400, 2867823842331030, 101430044142353688, 1466691026650416784, 7141303650101886, 615048947375476321, 1710023069407535280, 615005794121252401], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [2, 2, 6, 1], [2, 4, 3, 1], [2, 4, 6, 1], [4, 2, 2, 2], [4, 2, 6, 1], [4, 4, 2, 2], [4, 4, 3, 1], [4, 4, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^{10}.D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '1024.djt', 'autcent_hash': 1957374279625110839, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 8], [2, 4, 9], [4, 2, 10], [4, 4, 13]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '32.51', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 2220, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 8], [2, 4, 1, 9], [4, 2, 1, 8], [4, 2, 2, 1], [4, 4, 1, 13]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 833280, 'exponent': 4, 'exponents_of_order': [7], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '32.51', 'hash': 2220, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2, 2, 2, 1, 1], 'inner_gens': [[41455447, 125199496, 14, 41091256, 301245976, 132452760, 16], [171682687, 11290336, 5, 172046896, 95252536, 132452760, 16], [41455463, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 125199496, 5, 41091256, 95252536, 132452760, 16], [171682687, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 11290336, 5, 41091256, 95252536, 132452760, 16]], 'inner_hash': 51, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2^5', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 4]], 'label': '128.2220', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4^2:C2', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 44, 'number_divisions': 43, 'number_normal_subgroups': 430, 'number_subgroup_autclasses': 161, 'number_subgroup_classes': 829, 'number_subgroups': 1444, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 55], [4, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 6, 2, 2], 'outer_gen_pows': [95252536, 0, 0, 0, 0, 0, 0], 'outer_gens': [[207340583, 11290336, 5, 41091256, 95252536, 132452760, 16], [41455447, 11290336, 132452765, 41091256, 95252536, 132452760, 16], [12782303, 41091240, 5, 11290320, 95252536, 132452760, 16], [41455463, 11290320, 132452765, 41091240, 95252520, 132452760, 16], [12782303, 41091240, 132452774, 247258080, 95252536, 132452760, 16], [41455447, 11290320, 5, 41091256, 95252536, 132452760, 16], [41455463, 11290320, 5, 41091240, 95252536, 132452760, 16]], 'outer_group': '384.20162', 'outer_hash': 20162, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [5167, 11527, 368047, 23, 7257616, 87091200, 40279680], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 8], [4, 2], [8, 1]], 'representations': {'PC': {'code': 37156913670004904, 'gens': [1, 2, 3, 4, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 1430, 346, 80, 530, 124]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729084381808, 24992892284608672, 58438755861063890, 125055985569224674, 125055996420182666]}, 'Perm': {'d': 12, 'gens': [41455447, 11290336, 5, 41091256, 95252536, 132452760, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4^2:C_2', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
